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The derivative of the function/ x-1+(1+ln^2x/x)

Derivative of x-1+(1+ln^2x/x)

The solution

You have entered [src]

               2   
            log (x)
x - 1 + 1 + -------
               x

$$\left(1 + \frac{\log{\left(x \right)}^{2}}{x}\right) + \left(x - 1\right)$$

x - 1 + 1 + log(x)^2/x

Detail solution

Differentiate $\left(1 + \frac{\log{\left(x \right)}^{2}}{x}\right) + \left(x - 1\right)$ term by term:
1. Differentiate $x - 1$ term by term:
  1. Apply the power rule: $x$ goes to $1$
  2. The derivative of the constant $-1$ is zero.
  The result is: $1$
2. Differentiate $1 + \frac{\log{\left(x \right)}^{2}}{x}$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \log{\left(x \right)}^{2}$ and $g{\left(x \right)} = x$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = \log{\left(x \right)}$ .
    2. Apply the power rule: $u^{2}$ goes to $2 u$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
      1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      The result of the chain rule is:
      
      $\frac{2 \log{\left(x \right)}}{x}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Apply the power rule: $x$ goes to $1$
    Now plug in to the quotient rule:
    
    $\frac{- \log{\left(x \right)}^{2} + 2 \log{\left(x \right)}}{x^{2}}$
  The result is: $\frac{- \log{\left(x \right)}^{2} + 2 \log{\left(x \right)}}{x^{2}}$
The result is: $1 + \frac{- \log{\left(x \right)}^{2} + 2 \log{\left(x \right)}}{x^{2}}$
Now simplify:

$\frac{x^{2} - \log{\left(x \right)}^{2} + 2 \log{\left(x \right)}}{x^{2}}$

The answer is:

$\frac{x^{2} - \log{\left(x \right)}^{2} + 2 \log{\left(x \right)}}{x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2              
    log (x)   2*log(x)
1 - ------- + --------
        2         2   
       x         x

$$1 - \frac{\log{\left(x \right)}^{2}}{x^{2}} + \frac{2 \log{\left(x \right)}}{x^{2}}$$

Simplify

The second derivative [src]

  /       2              \
2*\1 + log (x) - 3*log(x)/
--------------------------
             3            
            x

$$\frac{2 \left(\log{\left(x \right)}^{2} - 3 \log{\left(x \right)} + 1\right)}{x^{3}}$$

Simplify

The third derivative [src]

  /          2               \
2*\-6 - 3*log (x) + 11*log(x)/
------------------------------
               4              
              x

$$\frac{2 \left(- 3 \log{\left(x \right)}^{2} + 11 \log{\left(x \right)} - 6\right)}{x^{4}}$$

Simplify

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Derivative of x-1+(1+ln^2x/x)

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The solution